# Prove that the series SUM (-1)^n n/p_n converges where p_n are primes

## Homework Statement

Prove that $$\sum(-1)^n\frac{n}{p_n}$$ converges, where $$p_n$$ is the nth prime.

## Homework Equations

The sequence $$\frac{n}{p_n}$$ is definately not monotone if there exists infinitely many twin primes, since $$2n-p_n<0$$ for sufficiently large n, so alternating series test is out. Are there any other ways of showing this converges?

$$\lim_{n \rightarrow \infty} \frac{p_n}{n \ln n} = 1$$